The Limit of a Function

1. Determine the limit of the following nine problems. solution

\(\displaystyle \lim_{x \to 1 } \, \dfrac{x^2-1}{x-1}\)   \(\displaystyle \lim_{x \to -1 } \, \dfrac{x^2-1}{x+1}\)  \(\displaystyle \lim_{x \to 2 } \, \dfrac{x^2-4}{x-2} \)  \(\displaystyle \lim_{x \to 1 } \, \dfrac{x^2-3x+2}{2x-2}\)

\(\displaystyle \lim_{x \to -2} \, \dfrac{x^2+5x+6}{x^2+2x}\)   \(\displaystyle \lim_{x \to 3 } \, \dfrac{x^2-5x+6}{x^2-9}\)   \(\displaystyle \lim_{x \to \infty} \, \dfrac{2x^3-6x^2+5}{6x^3-3x+5}\)

\(\displaystyle \lim_{x \to \infty} \, \dfrac{2x^5-6x^2+5}{3x^4-3x+5}\)   \(\displaystyle \lim_{x \to- \infty} \, \dfrac{5x^4-6x+15}{2x^5-x^2+7}\)

2. Determine the limit of the following five problems.  solution

\(\displaystyle \lim_{x \to \frac{1}{2}}\cos (\pi x) -e^{x-\frac{1}{2}}\) \(\displaystyle \lim_{x \to -1}\frac{x^2-x-2}{7x+7}\) \(\displaystyle \lim_{h \to 0}\frac{\sqrt{16+h}-4}{h}\)

\(\displaystyle \lim_{x \to 0}\dfrac{\frac{1}{x+4}-\frac{1}{4}}{x}\) \(\displaystyle \lim_{x \to \infty} \, \dfrac{3x^5+5x+2}{2x^5-2x^3+1}\)